1) State an equation satisfied by alpha, and hence show that alpha is the x coordinate of a point on the curve where y=3.
2) Use this iterative formula to find alpha correct to 2 decimal places, showing each iteration.
Solution: Assume that this sequence of value given by the iterative formula x(n+1)=2/9-ln[x(n)] with initial value x=1, converges to alpha, using letter 'a' to represent alpha. ie., x(n)->a as n->infinity.
Then, according to the iterative formula x(n+1)=2/9-ln[x(n)], we let n approach infinity, we have
So the limit of this sequence satisfy equation (1).
From (1) we can rewrite it as ...
This provides a sequence given by an iterative formula, and shows how to find an equation satisfied by the convergence alpha and use the formula to find alpha.